exponential integral

In mathematics, the exponential integral Ei(x) is defined as

\mbox{Ei}(x)=-\int_{-x}^{\infty} \frac{e^{-t}}{t} dt\,.

Since 1/t diverges at t=0, the above integral has to be understood in terms of the Cauchy principal value. The exponential integral arises also in the following sum:

\sum_{k=1}^{\infty} \frac{x^k}{k k!} = \mbox{Ei}(x)+\gamma+\ln x\,,

where γ is the Euler gamma constant. The exponential integral is closely related to the logarithmic integral function li(x),

li(x) = Ei (ln (x)) for all positive real x ≠ 1.

Also closely related is a function which integrates over a different range:

{\rm E}_1(x) = \int_x^\infty \frac{e^{-t}}{t} dt\,.

This function may be regarded as extending the exponential integral to the negative reals by

{\rm Ei}(-x) = - {\rm E}_1(x)

. We can express both of them in terms of an entire function,

{\rm Ein}(x) = \int_0^x (1-e^{-t})\frac{dt}{t}

= \sum_{k=1}^\infty \frac{(-1)^{k+1}x^k}{k k!}. Using this function, we then may define, using the logarithm,

{\rm E}_1(x) = -\gamma-\ln x + {\rm Ein}(x)

and

{\rm Ei}(x) = \gamma+\ln x + {\rm Ein}(-x)

. The exponential integral may also be generalized to

E_n(x) = \int_1^\infty \frac{e^{-xt}}{t} dt

References

Abromowitz, M., and Stegun, I. A., eds., Handbook of Mathematical Functions, fourth edition

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